Brief paperNoncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators☆
Introduction
It is well known that the small-gain theorem plays a fundamental role in robust stability analysis and -analysis (Packard & Doyle, 1993) is effective when we deal with structured uncertainties. The latter employs structured frequency-dependent scaling according to the structure of uncertainties, but it is cumbersome to get suitable frequency-dependent scaling (see, e.g., Zhou and Doyle (1998)). On the other hand, this article aims at investigating the use of the discrete-time lifting technique in robust stability analysis, and introduces a natural notion that accompanies such treatment, which we call noncausal linear periodically time-varying (LPTV) scaling.
In contrast to the conventional frequency-domain scaling, this new scaling can be regarded as a sort of time-domain scaling, and can be effective even for unstructured uncertainties. More importantly, we show that when the system is LTI, it can be regarded as inducing frequency-dependent scaling if it is interpreted in the conventional context in the original time axis without lifting.To put it another way, by using noncausal LPTV scaling, we can, in a sense, convert the search of frequency-dependent scaling into that of static scaling by the discrete-time lifting technique. In such a connection, it should be noted that noncausal LPTV scaling generalizes the conventional LTI scaling in two ways, i.e., (i) by introducing LPTV characteristics and (ii) by allowing noncausal operations in scaling. Regarding these two aspects, we show that (ii) is the most important source for the effectiveness of noncausal LPTV scaling in the frequency domain.
The contents of this article are as follows. We first describe the lifting-based robust stability analysis of discrete-time systems in Section 2. Based on the lifting-based treatment, we present in Section 3 the basic ideas for noncausal LPTV scaling. A numerical example with an LPTV nominal system is given there, which shows that noncausal LPTV scaling is indeed effective for robust stability analysis of discrete-time systems, and also motivates the subsequent discussions. In Section 4, we confine ourselves to the case with LTI systems, and give some theoretical results that suggest the effectiveness of noncausal LPTV scaling. In particular, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling if it is interpreted in the conventional lifting-free treatment. It is also shown that this promising property can be exploited even when the systems are LPTV rather than LTI, if we consider the -lifted transfer matrices of -periodic systems. Section 5 summarizes the arguments of the article and gives some remarks on further research directions.
Notation: denotes the set of positive integers.
Section snippets
Robust stability of discrete-time closed-loop systems
Consider the discrete-time system shown in Fig. 1 with the nominal system and the uncertainty . We assume that is a -input -output, internally stable, finite-dimensional (FD) LPTV system with period (i.e., an -periodic system), where . We assume that for some given set (possibly consisting of static systems only) satisfying the assumption:
A1 Every is FD, -periodic, internally stable, and is a connected set such that .
We sometimes consider the case when every
Noncausal LPTV scaling and a motivating numerical example
In this section, we give a brief idea on the technique that we introduce in this article, i.e., noncausal LPTV scaling. We consider a simple and special case of noncausal LPTV scaling, apply it to a numerical example, and demonstrate its effectiveness. The aim of this section, however, rather lies in motivating the theoretical discussions in this article.
Assuming here, let us consider a typical separator of the form corresponding to the
Properties of noncausal LPTV scaling applied to LTI systems
For the reason stated in the preceding section, we confine ourselves to the case when and are LTI in this section. Then and can be associated with their transfer matrices and , respectively, as well as their -lifted transfer matrices and for taken arbitrarily. Note that the forward shift in time is denoted by to distinguish it from the symbol for that in the lifted domain (i.e., corresponds to ). Obviously, Proposition 1 holds under the replacement of
Conclusion
This article studied the use of lifting in the robust stability analysis of discrete-time systems, which naturally leads to a novel technique called noncausal LPTV scaling. A numerical example with an LPTV nominal system was first studied to show the effectiveness of such a technique. We then discussed the properties of noncausal LPTV scaling applied to LTI systems, and showed that it has an important interpretation. That is, we showed that even static noncausal LPTV scaling induces
Tomomichi Hagiwara was born in Osaka, Japan on March 28, 1962. He received his B.E., M.E. and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan in 1984, 1986 and 1990, respectively. Since 1986, he has been with the Department of Electrical Engineering, Kyoto University, where he is Professor since 2001. His research interests include dynamical system theory and control theory such as analysis and design of sampled-data systems, time-delay systems and
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Tomomichi Hagiwara was born in Osaka, Japan on March 28, 1962. He received his B.E., M.E. and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan in 1984, 1986 and 1990, respectively. Since 1986, he has been with the Department of Electrical Engineering, Kyoto University, where he is Professor since 2001. His research interests include dynamical system theory and control theory such as analysis and design of sampled-data systems, time-delay systems and two-degree-of-freedom control systems.
Yasuhiro Ohara was born in Hyogo, Japan on January 28, 1985. He received his B.E. and M.E degrees in electrical engineering from Kyoto University, Kyoto, Japan in 2007 and 2009, respectively. Since 2009, he has been with Honda R&D Company, Tochigi, Japan.
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The material in this article was partially presented at the IFAC Workshop on PSYCO ’07, August 29–31, 2007, St. Petersburg and the IFAC World Congress 2008, July 6–11, 2008, Seoul. This article was recommended for publication in revised form by Associate Editor Lihua Xie under the direction of Editor Roberto Tempo.